A sequence of real numbers is equidistributed on an interval
if the probability of finding in any subinterval is proportional to the subinterval length.
The points of an equidistributed sequence form a dense
set on the interval .

However, dense sets need not necessarily be equidistributed. For example, , where is the fractional part,
is dense in but not equidistributed, as illustrated above for
to 5000 (left) and to (right)

Hardy and Littlewood (1914) proved that the sequence , of power
fractional parts is equidistributed for almost all real numbers (i.e., the exceptional set has Lebesguemeasure zero). Exceptional numbers include the positive
integers, the silver ratio (Finch 2003), and the golden
ratio .

The top set of above plots show the values of for equal to e, the Euler-Mascheroni
constant ,
the golden ratio , and pi. Similarly, the bottom set
of above plots show a histogram of the distribution of for these constants. Note that while
most settle down to a uniform-appearing distribution, curiously appears nonuniform after iterations. Steinhaus (1999) remarks that the highly uniform
distribution of has its roots in the form of the continued
fraction for .

Now consider the number of empty intervals in the distribution of in the intervals bounded by the intervals
determined by 0, , , ..., , 1 for , 2, ..., summarized below for the constants previously considered.